Changing Multiple Influences Simultaneously
You can set up a custom property and drivers to do do this.

Basically you do the following:


add a custom property to your armature to control all the influences (see)
For one of the copy rotation constraints, right click and select add driver->manually create later
Go to the graph editor view, put it in drivers mode and find the driver you just created and press n to show the drivers panel
Set its variable to a type of single property and select your armature as the object
Find your custom property and right-click on it and copy data path. Paste the data path into the path field on the driver variable.
update the Expr field in the driver panel to the name of your driver variable (by default this is var)
for each other constraint, you can right-click on the first constraint's influence and copy the driver, then use right click to paste to the new constraint
You can now update the custom property and all the constraint influences will be updated at the same time.